Piezoelectric quartz crystal resonators have long been employed to develop highly accurate timing signals for such applications as communications, navigation and radar. In particular, resonant frequencies of thickness shear mode quartz resonators are commonly used as timing standards in crystal-controlled oscillators.
In spite of the relative stability and precision of the frequency output of such quartz resonator controlled oscillators, frequency shifts and thus timing errors can occur when the resonator is subjected to acceleration (or gravity) caused stresses. These stresses are produced in the resonator as a result of interaction between the crystal element and its mounting or holding structure. Investigations have generally been unsuccessful in identifying effects, or whether there is any effect, of various parameters (crystal geometry, angle of cut, temperature, etc.) on acceleration sensitivity (see Filler et al "Further Studies on the Acceleration Sensitivity of Quartz Resonators", Proc. 37th Annual Symposium on Frequency Control, 1983, pp. 265-271; and Filler, Raymond L., "The Acceleration Sensitivity of Quartz Crystal Oscillators: A Review", Proc. 41st Annual Symposium on Frequency Control, 1987, pp. 398-408.) Vig et al, U.S. Pat. No. 4,451,755, does suggest some dependence of acceleration sensitivity on the radius of curvature of convex surfaces of a crystal. In a related co-pending patent application, EerNisse and Ward show that the position and/or shape of the active region of vibration of the crystal is an important factor in determining the acceleration sensitivity.
A resonator's sensitivity to acceleration has been defined by a so-called "gamma vector". The vector is composed of three frequency shift components which coincide with the x, y and z mechanical (or geometric) axes of the crystal and which are measured for acceleration applied in directions corresponding to each of the axes. Once the gamma vector is known for a crystal, the frequency shift for any acceleration vector a can be obtained as the dot product of that vector a and the gamma vector.
It is desirable, of course, to reduce the magnitude of the gamma vector as much as possible to reduce acceleration-caused frequency error. But, as indicated earlier, there has been little success in doing this in a practical and consistent manner. Crystals prepared and mounted in seemingly an identical fashion can, nevertheless, have different gamma vectors (in both direction and magnitude) for no apparent reason. The variation in gamma vector from resonator to resonator can be large or small; however, the more precise is the resonator fabrication and mounting geometry, the smaller is the vector deviation in magnitude and direction.
The earlier mentioned co-pending application showed that the gamma vector could be modified (normally reduced) by either moving the active region of vibration or changing its shape. This was done by adding or subtracting mass, or a combination of both, over selected areas of the resonator surface. In the case where the gamma vector changes significantly from resonator to resonator, it would be necessary to measure at least one component of the gamma vector in order to determine the required resonator mass modification. In the case of a tightly controlled manufacturing process where the gamma vectors are grouped closely together from resonator to resonator, measurement of the gamma vector for every resonator would be expensive for little additional benefit, since learning of the gamma vector for one resonator would provide a reasonable indication of the gamma vectors for the other similarly manufactured resonators.
It should be noted that crystal resonators used in frequency control are typically provided with their resonant frequency by "trimming" the resonators to preselected values. This trimming is carried out by adding mass uniformly over a crystal to lower the frequency to the desired level. Such addition of mass, however, has not been utilized to reduce the gamma vector of resonators.